The radius of the base of a cylinder is increasing at a rate of 7 millimeters per hour. The height of the cylinder is fixed at 1.5 millimeters. At a certain instant, the radius is 12 millimeters. What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)? Choose 1 answer: (A) 252 pi (B) 216 pi (C) 1512 pi (D) 126 pi The volume of a cylinder with radius r and height h is pir^(2)h.

Question

The radius of the base of a cylinder is increasing at a rate of 7 millimeters per hour. The height of the cylinder is fixed at 1.5 millimeters. At a certain instant, the radius is 12 millimeters. What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)? Choose 1 answer: (A)  252 pi (B)  216 pi (C)  1512 pi (D)  126 pi The volume of a cylinder with radius  r and height  h is  pir^(2)h.

Bytelearn · Accepted Answer

Volume Formula Derivation: The formula for the volume of a cylinder is V = πr^2h. We need to find dV/dt, the rate of change of volume with respect to time.Given Information: Given that dr/dt = 7 mm/hour (rate at which radius is increasing) and h = 1.5 mm (height of the cylinder is constant).Differentiate Volume Formula: Differentiate the volume formula with respect to time: dV/dt = d(πr^2h)/dt = πh * d(r^2)/dt.Differentiate r^2 with respect to time: Now differentiate r^2 with respect to time: d(r^2)/dt = 2r * dr/dt.Substitute Values: Substitute the values of dr/dt and r into the equation: d(r^2)/dt = 2 * 12 mm * 7 mm/hour = 168 mm^2/hour.Calculate dV/dt: Now substitute d(r^2)/dt and h into the dV/dt equation: dV/dt = π * 1.5 mm * 168 mm^2/hour.Calculate dV/dt: dV/dt = π * 1.5 * 168 = 252π mm^3/hour.

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